Solomon's store can be accessed through your extras menu or customization interface. Within the store you'll be shown new items, a list of categories of items available, and a handy little button to let you purchase all the RuneCoins you want. Items and animations can be previewed, so you'll be able to see precisely how awesome whatever you're buying is. In order to display these items, titles, and animations, you'll have to go into the customization interface of your equipment panel, where you'll be able to fine-tune details like what item is controlling your stats, what item sets your appearance, what title is on display, what color your outfits are, and more.
As a special treat, every RuneScape player can receive a one-time gift of 200 free RuneCoins through the store.
[Thanks to David for the tip!]